A Method of Successive Approximations
in the Framework of the Geometrized Lagrange Formalism
Grigory I. Garas’ko
Department of Physics, Scientific Research Institute of Hyper Complex Systems in Geometry and Physics Zavodskoy proezd 3, Frjazino, Moscow Region 141190, Russia
E-mail: gri9z@mail.ru
It is shown that in the weak field approximation the new geometrical approach can lead to the linear field equations for the several independent fields. For the stronger fields and in the second order approximation the field equations become non-linear, and the fields become dependent. This breaks the superposition principle for every separate field and produces the interaction between different fields.The unification of the gravitational and electromagnetic field theories is performed in frames of the geometrical approach in the pseudo-Riemannian space and in the curved Berwald-Moor space.
1 Introduction
In paper [1] the new (geometrical) approach was suggested for the field theory. It is applicable for any Finsler space [2] for which in any point of the main space x1; x2; : : : ; xn the indicatrix volume Vind(x1; x2; : : : ; xn)ev can be defined, provided the tangent space is Euclidean. Then the action I for the fields present in the metric function of the Finsler space is defined within the accuracy of a constant factor as a volume of a certainn-dimensional region V:
I =const Z
V
(n) dx1dx2: : : dxn
(Vind(x1; x2; : : : ; xn))ev:
(1)
Thus, the field Lagrangian is defined in the following way
L =const 1
(Vind(x1; x2; : : : ; xn))ev:
(2)
In papers [3, 4] the spaces conformally connected with the Minkowski space and with the Berwald-Moor space were regarded. These spaces have a single scalar field for which the field equation was written and the particular solutions were found for the spherical symmetry and for the rhombodode-caedron symmetry of the space.
The present paper is a continuation of those papers deal-ing with the study and development of the geometric field theory.
2 Pseudo-Riemannian space with the signature(+ )
Let us consider the pseudo-Riemannian space with the signa-ture (+ ) and select the Minkowski metric tensor gij in the metric tensor gij(x); of this space explicitly
gij(x) =gij +hij(x) : (3)
Let us suppose that the field hij(x) is weak, that is
jhij(x)j 1 : (4)
According to [1], the Lagrangian for a pseudo-Rieman-nian space with the signature (+ ) is equal to
L =
q
det(gij) : (5)
Let us calculate the value of [ det(gij)] within the ac-curacy of jhij(x)j2:
det(gij) ' 1 + L1+L2; (6) where
L1 =gijhij h00 h11 h22 h33; (7)
L2 = h00(h11+ h22+ h33) + h11h22+ h11h33+
+ h22h33 h212 h213 h223+ h203+ h202+ h201:
(8)
The last formula can be rewritten in a more conven-ient way
L2 = h00 h01
h01 h11
h00 h02
h02 h22
h00 h03
h03 h33
+ h11 h12
h12 h22
+
+ h11 h13
h13 h33
+ h22 h23
h23 h33
;
(9)
then
L ' 1 +1
2L1+ 1 2
L2 1
4L21
: (10)
To obtain the field equations in the first order approxi-mation, one should use the Lagrangian L1; and to do the same in the second order approximation — the Lagrangian
3 Scalar field
For the single scalar field '(x) the simplest representation of tensor hij(x) has the form :
hij(x) h(')ij (x) = @x@'i
@'
@xj: (11)
That is why
L' =
q
det(gij) = p1 L1' 1 1
2L1 1
8L21; (12)
where
L1 =
@' @x0
2 @'
@x1
2 @'
@x2
2 @'
@x3
2
: (13)
In the first order approximation, we can use the Lagran-gian L1 to obtain the following field equation
@2'
@x0@x0
@2'
@x1@x1
@2'
@x2@x2
@2'
@x3@x3 = 0 ; (14)
which presents the wave equation. The stationary field that depends only on the radius
r = p(x1)2+ (x2)2+ (x3)2; (15)
will satisfy the equation
d dr
r2d'
dr
= 0 ; (16)
the integration of which gives d'
dr = C1 1
r2 ) '(r) = C0+ C1
1
r: (17)
In the second order approximation one should use the La-grangian L1 14L21 to obtain the field equation in the sec-ond order approximation
gij @
@xi
1 1
2L1
@' @xj
= 0 ; (18)
this equation is already non-linear.
The strict field equation for the tensor hij(x) (11) is
gij @
@xi
0 B @
@' @xj
p 1 L1
1 C
A = 0 ; (19)
then the stationary field depending only on the radius must satisfy the equation
d dr
0 B B B B @r
2
d' dr s
1
d' dr
2
1 C C C C
A = 0 : (20)
Integrating it, we get d' dr =
C1
p
r4 C2 1
)
) '(r) = C0+ 1
Z
r
C1
p r4 C2
1
dr : (21)
The field with the upper sign and the field with the lower sign differ qualitatively: the upper sign “+” in Eq. (11) gives a finite field with no singularity in the whole space, the lower sign “ ” in Eq. (11) gives a field defined everywhere but for the spherical region
0 6r 6 p
jC1j ; (22)
in which there is no field, while
r > pjC1j ; r !
p
jC1j ) d'dr ! C1 1 : (23)
At the same time in the infinity (r ! 1) both solutions '(r) behave as the solution of the wave equation Eq. (17). If we know the Lagrangian, we can write the energy-momentum tensor Tji for the these solutions and calculate the energy of the system derived by the light speed c:
P0 =const Z (3)
T0
0dV : (24)
To obtain the stationary spherically symmetric solutions, we get
T0
0 = r
2
p r4 C2
1
; (25)
that is why for both upper and lower signs P0! 1. The metric tensor of Eq. (3,11) is the simplest way to “in-sert” the gravity field into the Minkowski space — the initial flat space containing no fields. Adding several such terms as in Eq. (11) to the metric tensor, we can describe more and more complicated fields by tensorhij= h(grav)ij :
4 Covariant vector field
To construct the twice covariant symmetric tensor hij(x)
with the help of a covariant field Ai(x) not using the con-nection objects, pay attention to the fact that the alternated partial derivative of a tensor is a tensor too
Fij = @A@xji @A@xji; (26)
but a skew-symmetric one. Let us construct the symmetric tensor on the base of tensor Fij. To do this, first, form a scalar
LA=gij gkmFikFjm =
= 2gij gkm@Ak
@xi
@Am
@xj
@Ak
@xi
@Aj
@xm
which gives the following expressions for two symmetric tensors
h(1)ij =gkm2@Ak
@xi
@Am
@xj
@Ak
@xi
@Aj
@xm
@Ak
@xj
@Ai
@xm
; (28)
h(2)ij =gkm2@Ai
@xk
@Aj
@xm
@Ai
@xk
@Am
@xj
@Aj
@xk
@Am
@xi
: (29)
Notice, that not only Fij and LA but also the tensors h(1)ij , h(2)ij are gradient invariant, that is they don’t change with transformations
Ai ! Ai+@f(x)@xi ; (30)
where f(x) is an arbitrary scalar function. Let
hij h(Aijk) = (x) h(1)ij + [1 (x)] h(2)ij ; (31)
where (x) is some scalar function. Then in the first order approximation we get
L1 = 2gij gkm
Ak
@xi
@Aj
@xm
LA; (32)
and the first order approximation for the field Ai(x) gives Maxwell equations
gij @2
@xi@xjAk
@ @xk
gij @Aj
@xi
= 0 : (33)
For Lorentz gauge
gij @Aj
@xi = 0 ; (34)
the equations Eqs. (33) take the form
Ak = 0 : (35)
It is possible that Eq. (31) is not the most general form for tensor hij which in the first order approximation gives the field equations coinciding with Maxwell equations.
To obtain Maxwell equations not for the free field but for the field with sources ji(x); one should add to h(Aijk) Eq. (31) the following tensor
h(jk)
ij =
16
c
12(Aijj+ Ajji) : (36)
This means that the metric tensor Eq. (3) with tensor
hij = h(Max)ij h(Aijk)+ h(jijk) (37)
describes the weak electromagnetic field with source jk(x): We must bear in mind that we use the geometrical approach to the field theory, and we have to considerjk(x) to be given and not obtained from the field equations.
So, the metric tensor Eq. (3) with tensor
hij = h(Aijk)+ h(grav)ij ; (38)
where ; are the fundamental constants in frames of the unique pseudo-Riemannian geometry describes simultane-ously the free electromagnetic field and the free gravitational field. To include the sources, jk(x); of the electromagnetic field, the metric tensor must either include not onlyjk(x)but the partial derivatives of this field too or the fieldjk(x) must be expressed by the other fields as shown below.
If the gravity field is “inserted” in the simplest way as shown in the previous section, then the sources of the electro-magnetic field can be expressed by the scalar field as follows
ji(x) = q@x@'i : (39)
In this case the first order approximation for Lorentz gauge gives
Ak =4
c jk; (40)
' = 0 : (41)
Since the density of the current has the form of Eq. (39), the Eq. (41) gives the continuity equation
gij@ji
@xj = 0 : (42)
5 Several weak fields
The transition from the weak fields to the strong fields may lead to the transition from the linear equations for the inde-pendent fields to the non-linear field equations for the mutu-ally dependent interacting fields'(x) and (x)“including” gravity field in the Minkowski space.
Let
hij = "'@x@'i@x@'j + " @x@ i@x@ j ; (43)
where "'," are independent sign coefficients. Then the strict Lagrangian can be written as follows
L'; =
q
1 +L1+L2; (44)
and
L1 =gij
"'@x@'i@x@'j + " @x@ i@x@ j
; (45)
L2= "'" " @' @x0 @ @x1 @' @x1 @ @x0 2 @' @x0 @ @x2 @' @x2 @ @x0 2 @' @x0 @ @x3 @' @x3 @ @x0 2 + + @' @x1 @ @x2 @' @x2 @ @x1 2 + + @' @x1 @ @x3 @' @x3 @ @x1 2 + + @' @x2 @ @x3 @' @x3 @ @x2 2# 9 > > > > > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > > > > > ; : (46)
This formula, Eq. (46), can be obtained from Eq. (9) most easily, if one uses the following simplifying formula
hii hi j
hi j hjj
= @' @xi @ @xi @' @xj @ @xj 2 = = @' @xi @ @xj @' @xj @ @xi 2 :
In the first order approximation for the Lagrangian, the expression L1 should be used. Then the field equations give the system of two independent wave equations
@2'
@x0@x0
@2'
@x1@x1
@2'
@x2@x2
@2'
@x3@x3 = 0
@2
@x0@x0
@2
@x1@x1
@2
@x2@x2
@2
@x3@x3 = 0
9 > > > = > > > ; :
Here the fields '(x) and (x) are independent and the superposition principle is fulfilled.
Using the strict Lagrangian for the two scalar fields Eq. (44) we get a system of two non-linear differential equa-tions of the second order
gij @
@xi
2 4';j
1 grs ;r ;s ;j grs';r ;s
p
1 +L1+L2
3 5 = 0 ;
gij @
@xi
2 4 ;j
1+grs' ;r';s
';j grs';r ;s
p
1 +L1+L2
3 5 = 0 ;
where comma means the partial derivative. Here the fields '(x), (x)depend on each other, and the superposition prin-ciple is not fulfilled. The transition from the last but one equa-tions to the last equaequa-tions may be regarded as the transition from the weak fields to the strong fields.
6 Non-degenerate polynumbers
Consider a certain system of the non-degenerate polynum-bers Pn [5], that isn-dimensional associative commutative non-degenerated hyper complex numbers. The correspond-ing coordinate space x1; x2; : : : ; xn is a Finsler metric flat space with the length element equal to
ds = n q
gi1i2:::in dxi1dxi2: : : dxin; (47)
where gi1i2:::in is the metric tensor which does not depend on the point of the space. The Finsler spaces of this kind can be found in literature (e.g. [6, 7, 8, 9]) but the fact that all the non-degenerated polynumber spaces belong to this type of Finsler spaces was established beginning from the papers [10, 11] and the subsequent papers of the same authors, es-pecially in [5].
The components of the generalized momentum in geom-etry corresponding to Eq. (47) can be found by the formulas
pi =
gij2:::jn dxj2: : : dxjn
gi1i2:::in dxi1dxi2: : : dxin n 1
n :
(48)
The tangent equation of the indicatrix in the space of the non-degenerated polynumbers Pn can be always written [5] as follows
gi1i2:::inp
i1pi2: : : pin n = 0 ; (49)
where is a constant. There always can be found such a basis (and even several such bases) and such a > 0 that
gi1i2:::in =g
i1i2:::in
: (50)
Let us pass to a new Finsler geometry on the base of the space of non-degenerated polynumbers Pn. This new geom-etry is not flat, but its difference from the initial geomgeom-etry is infinitely small, and the length element in this new geome-try is
ds = rhn g
i1i2:::in + "hi1i2:::in(x) i
dxi1dxi2: : : dxin; (51)
where" is an infinitely small value. If in the initial space the volume element was defined by the formula
dV = dxi1dxi2: : : dxin; (52)
in the new space within the accuracy of " in the first power we have
dVh '
h
1 + " C0gi1i2:::inhi1i2:::in(x) i
dxi1dxi2: : : dxin:
approximation is
L1 =gi1i2:::inh
i1i2:::in(x) : (53)
This expression generalizes formula Eq. (7).
7 Hyper complex spaceH4
In the physical (“orthonormal” [5]) basis in which every point of the space is characterized by the four real coordi-nates x0; x1, x2; x3 the fourth power of the length element dsH4 is defined by the formula
(dsH4)4
gijkldx0dx1dx2dx3 =
= (dx0+ dx1+ dx2+ dx3)(dx0+ dx1 dx2 dx3)
(dx0 dx1+ dx2 dx3)(dx0 dx1 dx2+ dx3) =
= (dx0)4+(dx1)4+(dx2)4+(dx3)4+8dx0dx1dx2dx3
2(dx0)2(dx1)2 2(dx0)2(dx2)2 2(dx0)2(dx3)2
2(dx1)2(dx2)2 2(dx1)2(dx3)2 2(dx2)2(dx3)2: (54)
Let us compare the fourth power of the length element dsH4 in the space of polynumbers H4 with the fourth power
of the length element dsMin in the Minkowski space
(dsMin)4= (dx0)4+(dx1)4+(dx2)4+(dx3)4 2(dx0)2(dx1)2 2(dx0)2(dx2)2 2(dx0)2(dx3)2
+ 2(dx1)2(dx2)2+2(dx1)2(dx3)2+2(dx2)2(dx3)2:
(55)
This means
(dsH4)4 = (dsMin)4+ 8dx0dx1dx2dx3
4(dx1)2(dx2)2 4(dx1)2(dx3)2 4(dx2)2(dx3)2; (56)
and in the covariant notation we have
(dsH4)4 =
gijgkl+13 g0
ijkl Gijkl
dxidxjdxkdxl;
(57)
where
g0 ijkl =
1 ; ifi; j; k; lare all different
0 ; else (58)
Gijkl=
8 > > < > > :
1 ; if i; j; k; l,0 and i = j,k = l;
or i = k,j = l;
or i = l,j = k 0 ; else
(59)
The tangent equation of the indicatrix in the H4 space can be written in the physical basis as in [5] :
(p0+ p1+ p2+ p3)(p0+ p1 p2 p3)
(p0 p1+ p2 p3)(p0 p1 p2+ p3) 1 = 0 ;
(60)
wherepiare the generalized momenta
pi =@ ds@(dxH4i): (61)
Comparing formula Eq. (60) with formula Eq. (61), we have
gijklp
ipjpkpl 1 = 0 : (62)
Here
gijkl =gij gkl+ 1
3
g0 ijkl Gijkl; (63)
and
gijkl =g ijkl
g0 ijkl =g0 ijkl
Gijkl =Gijkl
9 > > > > > = > > > > > ;
: (64)
To get the Lagrangian for the weak field in the first order approximation, we have to get tensorhijkl in Eq. (53). In the simplified version it could be splitted into two additive parts: gravitational part and electromagnetic part. The gravitational part can be constructed analogously to Sections 3 and 5 with regard to the possibility to use the two-index number tensors, since now tensors gijkl; hijkl have four indices. The con-struction of the electromagnetic part should be regarded in more detail.
Since we would like to preserve the gradient invariance of the Lagrangian and to get Maxwell equations for the free field in theH4 space, let us write the electromagnetic part of tensorhijkl in the following way
hAk
ijkl = (x) h(1)ijkl+ [1 (x)] h(2)ijkl; (65)
where the tensors h(1)ijkl; h(2)ijkl are the tensors present in the round brackets in the r.h.s. of formulas Eqs. (28,29). Then
LA = gijklhAk
ijkl
gij gkm@Ak
@xi
@Am
@xj
@Ak
@xi
@Aj
@xm
:
(66)
To obtain Maxwell equations not for the free field but for the field with a source ji(x); one should add to the tensor h(Ak)
ijkl Eq. (65) the following tensor
h(jk)
ijkl =
8
3 2Aijj
gkl Aigjkjl ji gjkAl;
;
symmetrized in all indices, that is tensor
describes the weak electromagnetic field with the sources ji(x), where
ji =
X
b
q(a)@ @x(b)i ; (67)
and (b) are the scalar components of the gravitational field. To obtain the unified theory for the gravitational and elec-tromagnetic fields one should take the linear combination of tensor h(Max)ijkl corresponding to the electromagnetic field in the first order approximation, and tensor h(grav)ijkl correspond-ing to the gravitational field in the first order approximation
hijkl = h(Max)ijkl + h(grav)ijkl ; (68)
where ; are constants. Tensor h(grav)ijkl may be, for exam-ple, constructed in the following way
hgravijkl =XN
a=1
"(a)@'@x(a)i @'@x(a)j @'@x(a)k @'@x(a)l +
+
M
X
b=1
(b)@ @x(b)(i @ @x(b)j gkl);
(69)
where "(a); (b) are the sign coefficients, and '(a); (b) are the scalar fields. The whole number of scalar fields is equal to (N + M).
8 Conclusion
In this paper it was shown that the geometrical approach [1] to the field theory in which there usually appear the non-linear and non-splitting field equations could give a system of inde-pendent linear equations for the weak fields in the first order approximation. When the fields become stronger the super-position principle (linearity) breaks, the equations become non-linear and the fields start to interact with each other. We may think that these changes of the equations that take place when we pass from the weak fields to the strong fields are due to the two mechanisms: first is the qualitative change of the field equations for the free fields in the first order approxima-tion; second is the appearance of the additional field sources, that is the generation of the field by the other fields.
In frames of the geometrical approach to the field theory [1] the unification of the electromagnetic and gravitational fields is performed both for the four-dimensional pseudo-Rie-mannian space with metric tensor gij(x) and for the four-dimensional curved Berwald-Moor space with metric tensor gijkl(x):
Acknowledgements
This work was supported in part by the Russian Foundation for Basic Research under grant RF BR 07 01 91681 RAa and by the Non-Commercial Foundation for Finsler Geometry Research.
Submitted on August 03, 2008/Accepted on August 11, 2008
References
1. Garas’ko G.I. Field theory and Finsler spaces.Hyper Complex
Numbers in Geometry and Physics, 2006, v. 3 (2), 6–20.
2. Rashevsky P. K. Geometrical theory of equations with partial derivatives. OGIZ, Moscow-Leningrad, 1947.
3. Garas’ko G.I. Particular stationary solution of the field equation for the space conformally connected with Minkowski space.
Hyper Complex Numbers in Geometry and Physics, 2007,
v. 4 (1), 26–37.
4. Garas’ko G.I. The space conformally connected with Berwald-Moor space. Hyper Complex Numbers in Geometry and
Physics, 2007, v. 4 (1), 38–51.
5. Pavlov D.G., Garas’ko G.I. Geometry of the non-degenerated polynumbers. Hyper Complex Numbers in Geometry and
Physics, 2007, v. 4 (1), 3–25.
6. Matsumoto M., Numata S. On Finsler spaces with 1-form
met-ric.Tensor, 1978, v. 32, 161–169.
7. Matsumoto M., Numata S. On Finsler spaces with 1-form met-ric II. Berwald — Moor’s metmet-ricL = (y1y2: : : yn)n1.Tensor, 1978, v. 32, 275–277.
8. Matsumoto M., Numata S. On Finsler spaces with a cubic
met-ric,Tensor, 1979, v. 33, 153–162.
9. Shimada H. On Finsler spaces with the metricLofm-th root metric.Tensor, 1979, v. 33, 365–372.
10. Pavlov D.G. Generalized axioms of the scalar product.
Hy-per Complex Numbers in Geometry and Physics, 2004, v. 1 (1),
6–20.
11. Garas’ko G.I. Generalized analytical functions of the polynum-ber variable. Hyper Complex Numbers in Geometry and
